Minimal Hilbert function for double schemes supported on linear configurations

Let $t$ and $r$ be integers with $0<r<t$, and consider double schemes in the projective plane whose support has the generic Hilbert function $(1,2,\dots,t-1,r)$. Let $C_{t,r}$ denote the corresponding linear configuration of points.

Minimal-Hilbert-function conjecture. Among these double schemes, there is a minimal Hilbert function, and it occurs when the support is $C_{t,r}$.

The claim concerns first infinitesimal neighborhoods of point configurations and extends the authors' computations for low-degree examples. They explicitly do not claim that $C_{t,r}$ is the unique support attaining the minimum; for example, the same Hilbert function can arise from another configuration when $(t,r)=(4,2)$.